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人気のある三角関数 >

証明する tan^2(x)=(1+tan^2(x))/(1+cot^2(x))

解

証明する

tan^{2} (x) = \frac{1 + tan ^{2} ( x )}{1 + cot ^{2} ( x )}

解

真

解答ステップ

tan^{2} (x) = \frac{1 + tan ^{2} ( x )}{1 + cot ^{2} ( x )}

右側を操作する

\frac{1 + tan ^{2} ( x )}{1 + cot ^{2} ( x )}

サイン, コサインで表わす

\frac{1 + tan ^{2} ( x )}{1 + cot ^{2} ( x )}

基本的な三角関数の公式を使用する:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{1 + cot ^{2} ( x )}

基本的な三角関数の公式を使用する:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

簡素化

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} : \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

結合

1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} : \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

元を分数に変換する:

1 = \frac{1 sin ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1 \cdot sin ^{2} ( x )}{sin ^{2} ( x )} + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

乗算:

1 \cdot sin^{2} (x) = sin^{2} (x)

= \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{\frac{s i n ^{2} ( x ) + c o s ^{2} ( x )}{s i n ^{2} ( x )}}

結合

1 + \frac{sin ^{2} ( x )}{cos ^{2} ( x )} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

1 + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

元を分数に変換する:

1 = \frac{1 cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{1 \cdot cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

乗算:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{c o s ^{2} ( x )}}{\frac{s i n ^{2} ( x ) + c o s ^{2} ( x )}{s i n ^{2} ( x )}}

分数を割る:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) sin ^{2} ( x )}{cos ^{2} ( x ) ( sin ^{2} ( x ) + cos ^{2} ( x ) )}

共通因数を約分する:

cos^{2} (x) + sin^{2} (x)

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

三角関数の公式を使用して書き換える

= \frac{sin ( x )}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )}

基本的な三角関数の公式を使用する:

\frac{sin ( x )}{cos ( x )} = tan (x)

\frac{sin ( x ) tan ( x )}{cos ( x )}

= tan (x) \frac{sin ( x )}{cos ( x )}

基本的な三角関数の公式を使用する:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) tan (x)

簡素化

tan (x) tan (x) : tan^{2} (x)

tan (x) tan (x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

tan (x) tan (x) = tan^{1 + 1} (x)

= tan^{1 + 1} (x)

数を足す:

1 + 1 = 2

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

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p ro v e \frac{( tan ^{2} ( θ ) )}{( sec ( θ ) - 1 )} - 1 = sec (θ)

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p ro v e 2 sin^{2} (x) + cos (x) = 1

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p ro v e cos (x) + sin (x) tan^{2} (x) = sec (x)

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p ro v e csc^{2} (x) - cot^{2} (x) + 1 = 0

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p ro v e cos (- b) tan (b) csc (b) cot (b) = cot (b)